Place honeydew in blender, and process until smooth; pour into a bowl. Place cantaloupe in blender, and process until smooth; pour into a bowl. To each bowl of puréed melon, add 2 tablespoons vodka, 2 tablespoons sugar, and 2 teaspoons lime juice; stir well. Cover and chill. Place strawberries in blender; process until smooth. Pour into a bowl; cover and chill.
To serve, pour 1/2 cup of cantaloupe mixture into each of 4 individual bowls; pour 1/2 cup of honeydew mixture in center of cantaloupe mixture. Dollop each serving with 2 tablespoons of puréed strawberries; swirl decoratively with a wooden pick.